Moved into a new home in 2021. Does my Energy Tax Credit with the old house ($133 in 2020) follow me to the new house?

The limitations in amounts of the energy credits are by taxpayer and not by house.

I don’t quite understand your question.  If you claimed a credit in 2020 when you owned the house, and you have a carry forward because of your tax situation, you still get to use the carryforward even though you moved out of the house (because the original claim occurred when you lived in the house).

If that was not your question, please post a clarification.

Or, you might have been thinking about the $500 lifetime limitation for certain energy efficient building improvements like doors, windows, and furnaces and air conditioners. That credit goes with the taxpayer and not the house and is a lifetime cap.